Heya,
At over 1/2000s, you shouldn't have issues with blur without an extremely close range high magnification shot happening unless you're simply out of focus. My guess is that it's out of focus and not motion blur.
I use an old 7D classic. I use back button focus, AV mode (select my aperture and I dial in my own ISO and meter based on a minimum shutter speed based on the light I'm shooting in, with a goal of around 1/1000s or faster typically, but I do look to be around 1/1600s or 1/2000s, over 1/2000s is generally not necessary for me, but it can be for certain situations), spot focus (I don't use zones and I don't use multiple points when tracking a bird), I set my AI servo sensitivity lower (last notch closest to the lowest slowest setting), I prefer partial metering or spot metering and I generally don't use more than +1/3rd FEC with my 7D, but some other cameras I needed more to get the exposure I prefer. Some of these features may cross over to your camera.
But regardless of that, technique is a big deal too. When shooting with a long lens, especially hand held, it takes a lot more effort to pan with a bird at long focal lengths and at close range, while focusing, and composing a good pleasing composition before taking shots. I also take my shots in short bursts, I don't just spray & pray, I take bursts of one or two or even three shots at a time during the moment I'm trying to capture the wings doing something specific.
I shoot with both a 150-600 as well as a 300mm prime. I prefer the 300mm prime for in flight and prefer to simply be closer to targets with it (on APS-C; so equivalent for a full frame would be a 500mm prime). With my 150-600, I generally like to point it at stationary stuff, where I'm trying to reach out and can't get closer. I stress being closer because I shoot near water a lot, and the more environment and atmosphere between you and your target, the more potential for things to play against you.
Note, I turn off VC on my 150-600 when shooting in flight.
I like shooting between 300mm & 500mm on APS-C (or 500mm~600mm on full frame, as I don't have 800mm).
+++
Here's my Tamron 150-600 at 500mm on APS-C, which is common for me when chasing birds that I cannot get close to (like Osprey). I shot this with AV mode, F7.1 (just stopped down a wee bit), enough ISO to get close to 1/2000s (in my case here it was ISO 1000), using spot focus, partial metering, and on my 7D I leave my FEC around 0 or +1/3rd based on it's behavior.
IMAGE LINK: https://flic.kr/p/GqAgHU
IMG_1665
by
Martin Wise
, on Flickr
Otherwise, here's my 300mm F4L that I use more often for in flight and low light as I prefer a smaller, shorter lens on APS-C as it's much easier to control and pan for me than my full frame and a much larger heavier lens (and I just prefer to get closer anyways):
IMAGE LINK: https://flic.kr/p/Kfwt5X
IMG_2963
by
Martin Wise
, on Flickr
IMAGE LINK: https://flic.kr/p/Kfv7nB
IMG_3047
by
Martin Wise
, on Flickr
IMAGE LINK: https://flic.kr/p/Kbwd1c
IMG_2762
by
Martin Wise
, on Flickr
IMAGE LINK: https://flic.kr/p/KBVwKt
IMG_2417
by
Martin Wise
, on Flickr
I shoot a lot of smaller birds, so I'm often getting closer and I'm almost always near water (like 99% of the time if I'm not doing song birds at my blind), so I prefer shorter, lighter stuff (I also do this from a kayak, so it shapes my preferences).
Very best,